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Simplifying -5n(-3 + 5n) + 3n = -21 + -4n (-3 * -5n + 5n * -5n) + 3n = -21 + -4n (15n + -25n2) + 3n = -21 + -4n Reorder the terms: 15n + 3n + -25n2 = -21 + -4n Combine like terms: 15n + 3n = 18n 18n + -25n2 = -21 + -4n Solving 18n + -25n2 = -21 + -4n Solving for variable 'n'. Reorder the terms: 21 + 18n + 4n + -25n2 = -21 + -4n + 21 + 4n Combine like terms: 18n + 4n = 22n 21 + 22n + -25n2 = -21 + -4n + 21 + 4n Reorder the terms: 21 + 22n + -25n2 = -21 + 21 + -4n + 4n Combine like terms: -21 + 21 = 0 21 + 22n + -25n2 = 0 + -4n + 4n 21 + 22n + -25n2 = -4n + 4n Combine like terms: -4n + 4n = 0 21 + 22n + -25n2 = 0 Begin completing the square. Divide all terms by -25 the coefficient of the squared term: Divide each side by '-25'. -0.84 + -0.88n + n2 = 0 Move the constant term to the right: Add '0.84' to each side of the equation. -0.84 + -0.88n + 0.84 + n2 = 0 + 0.84 Reorder the terms: -0.84 + 0.84 + -0.88n + n2 = 0 + 0.84 Combine like terms: -0.84 + 0.84 = 0.00 0.00 + -0.88n + n2 = 0 + 0.84 -0.88n + n2 = 0 + 0.84 Combine like terms: 0 + 0.84 = 0.84 -0.88n + n2 = 0.84 The n term is -0.88n. Take half its coefficient (-0.44). Square it (0.1936) and add it to both sides. Add '0.1936' to each side of the equation. -0.88n + 0.1936 + n2 = 0.84 + 0.1936 Reorder the terms: 0.1936 + -0.88n + n2 = 0.84 + 0.1936 Combine like terms: 0.84 + 0.1936 = 1.0336 0.1936 + -0.88n + n2 = 1.0336 Factor a perfect square on the left side: (n + -0.44)(n + -0.44) = 1.0336 Calculate the square root of the right side: 1.016661202 Break this problem into two subproblems by setting (n + -0.44) equal to 1.016661202 and -1.016661202.Subproblem 1
n + -0.44 = 1.016661202 Simplifying n + -0.44 = 1.016661202 Reorder the terms: -0.44 + n = 1.016661202 Solving -0.44 + n = 1.016661202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.44' to each side of the equation. -0.44 + 0.44 + n = 1.016661202 + 0.44 Combine like terms: -0.44 + 0.44 = 0.00 0.00 + n = 1.016661202 + 0.44 n = 1.016661202 + 0.44 Combine like terms: 1.016661202 + 0.44 = 1.456661202 n = 1.456661202 Simplifying n = 1.456661202Subproblem 2
n + -0.44 = -1.016661202 Simplifying n + -0.44 = -1.016661202 Reorder the terms: -0.44 + n = -1.016661202 Solving -0.44 + n = -1.016661202 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.44' to each side of the equation. -0.44 + 0.44 + n = -1.016661202 + 0.44 Combine like terms: -0.44 + 0.44 = 0.00 0.00 + n = -1.016661202 + 0.44 n = -1.016661202 + 0.44 Combine like terms: -1.016661202 + 0.44 = -0.576661202 n = -0.576661202 Simplifying n = -0.576661202Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.456661202, -0.576661202}
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